Non-negative solutions of a sublinear elliptic problem
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2024
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| _version_ | 1866909131247452160 |
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| author | López-Gómez, Julián Rabinowitz, Paul H. Zanolin, Fabio |
| author_facet | López-Gómez, Julián Rabinowitz, Paul H. Zanolin, Fabio |
| contents | In this paper the existence of solutions, $(λ,u)$, of the problem $$-Δu=λu -a(x)|u|^{p-1}u \quad \hbox{in }Ω, \qquad
u=0 \quad \hbox{on}\;\;\partialΩ,$$ is explored for $0 < p < 1$. When $p>1$, it is known that there is an unbounded component of such solutions bifurcating from $(σ_1, 0)$, where $σ_1$ is the smallest eigenvalue of $-Δ$ in $Ω$ under Dirichlet boundary conditions on $\partialΩ$. These solutions have $u \in P$, the interior of the positive cone. The continuation argument used when $p>1$ to keep $u \in P$ fails if $0 < p < 1$. Nevertheless when $0 < p < 1$, we are still able to show that there is a component of solutions bifurcating from $(σ_1, \infty)$, unbounded outside of a neighborhood of $(σ_1, \infty)$, and having $u \gneq 0$. This non-negativity for $u$ cannot be improved as is shown via a detailed analysis of the simplest autonomous one-dimensional version of the problem: its set of non-negative solutions possesses a countable set of components, each of them consisting of positive solutions with a fixed (arbitrary) number of bumps. Finally, the structure of these components is fully described. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04396 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-negative solutions of a sublinear elliptic problem López-Gómez, Julián Rabinowitz, Paul H. Zanolin, Fabio Analysis of PDEs 35B09, 35B25, 35B32, 35J15 In this paper the existence of solutions, $(λ,u)$, of the problem $$-Δu=λu -a(x)|u|^{p-1}u \quad \hbox{in }Ω, \qquad u=0 \quad \hbox{on}\;\;\partialΩ,$$ is explored for $0 < p < 1$. When $p>1$, it is known that there is an unbounded component of such solutions bifurcating from $(σ_1, 0)$, where $σ_1$ is the smallest eigenvalue of $-Δ$ in $Ω$ under Dirichlet boundary conditions on $\partialΩ$. These solutions have $u \in P$, the interior of the positive cone. The continuation argument used when $p>1$ to keep $u \in P$ fails if $0 < p < 1$. Nevertheless when $0 < p < 1$, we are still able to show that there is a component of solutions bifurcating from $(σ_1, \infty)$, unbounded outside of a neighborhood of $(σ_1, \infty)$, and having $u \gneq 0$. This non-negativity for $u$ cannot be improved as is shown via a detailed analysis of the simplest autonomous one-dimensional version of the problem: its set of non-negative solutions possesses a countable set of components, each of them consisting of positive solutions with a fixed (arbitrary) number of bumps. Finally, the structure of these components is fully described. |
| title | Non-negative solutions of a sublinear elliptic problem |
| topic | Analysis of PDEs 35B09, 35B25, 35B32, 35J15 |
| url | https://arxiv.org/abs/2403.04396 |